Apparatus and test device for the application and measurement of prescribed, predicted and controlled contact pressure on wires

ABSTRACT

The mechanical behavior of wires subjected to axial loading and experiencing bending deformation is used to ensure effective control of the contact pressure in mechanical and/or heat removing devices, and similar structures and systems. An apparatus for taking advantage of the characteristics of wires in packaging of a device, such as a semiconductor device, is disclosed, as well as a test device for identifying the accurate contact pressure required in same. Methods for the prediction of such a behavior for pre-buckling, buckling, and post-buckling conditions in wires, carbon nanotubes (CNTs), and similar wire-grid-array (WGA) structures, for example are also disclosed.

CROSS REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. provisional patentapplication No. 60/615,601 filed Oct. 5, 2004, which application isincorporated herein in its entirety by the reference thereto

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention pertains generally to an apparatus and method for devicedesign, prediction of its mechanical behavior and passive control ofcontact pressure using the phenomena of bending of wires subjected toaxial loading, including the phenomenon of elastic stability, and with aparticular emphasis on mechanical, electrical, optical, and heatremoving devices.

2. Description of the Prior Art

The performance of many mechanical, electrical, optical and thermaldevices is sensitive to the level of, and the changes in, contactpressure. For example, a flexible (“Euler”, “Cobra”, buckled beam)electrical test probe provides an attractive solution to makingelectrical contact to a large number of close-center probe pads inadvanced packages for very large scale integration (VLSI) devices. Thepressure provided by such a probe should be high enough to ensuresufficient electrical contact, but low enough not to crack the chip orto cause damage to the probe itself. A similar situation takes place insome fiber-optic based devices, such as, optical sensors, opticalconnectors, as well as in some heat removing devices. The producedpressure caused by the bent and/or buckled heat conducting wires orcarbon nanotubes (CNTs) should be high enough for a satisfactory heattransfer performance of the thermal interface, but low enough not tocause damage to the hot body, which might be mechanically vulnerable,nor to the wires or the CNTs themselves. This requirement often imposessignificant restrictions on, and difficulties in, how thewire-grid-array (WGA) is designed, manufactured, and operated. In someapplications, such as, CNT-based advanced heat-sinks, it is practicallyimpossible to create a viable and functionally reliable product, ifeffective and insightful predictive modeling, preferably based on ananalytical approach, is introduced and carried out prior to actualdesign and manufacturing efforts.

Prior art solutions suffer from several deficiencies. Notably, severalmechanical and physical design problems arise in connection with such anapplication, including:

-   -   a) Insufficient anchoring strength of the root portion of the        rods, including weak adhesion to the base; and    -   b) Extraordinary variability of the diameters and lengths of the        grown wires and, because of that, a significant variability and        even uncertainty in the produced contact pressure.

Accordingly, it would be advantageous to develop a device, a designmethodology and practical solutions to enable one to realisticallyprescribe, thoroughly predict and effectively control the contactpressure. It would be further advantageous to examine different designoptions and develop simple and easy-to-use formulae and calculationprocedures that enable a WGA designer to choose the appropriatematerials and the adequate geometric characteristics of the WGA wires,and predict with high accuracy their mechanical behavior, includingcontact pressure, displacements, stresses and even the probability offunctional and/or mechanical failure.

SUMMARY OF THE INVENTION

The mechanical behavior of wires subjected to axial loading andexperiencing bending deformation is used to ensure effective control ofthe contact pressure in mechanical and/or heat removing devices, andsimilar structures and systems. An apparatus for taking advantage of thecharacteristics of wires in packaging of a device, such as asemiconductor device, is disclosed, as well as a test device foridentifying the accurate contact pressure required in same. Methods forthe prediction of such a behavior in the pre-buckling, buckling, andpost-buckling conditions in wires, carbon nanotubes (CNTs), and similarwire-grid-array (WGA) structures are also disclosed.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a cross-sectional view of an exemplary embodiment of a devicewith a WGA that could be bent or buckled;

FIG. 2 is a schematic diagram of a wire experiencing large elasticdeformations;

FIG. 3 is a table describing the post-buckling behavior of a wire andshowing results for the ratio of actual compressive force to critical(Euler) value of this force, the ratio of the lateral displacement(lateral coordinate) of a wire's end to the initial (non-deformed)length of the wire, and the ratio of the axial displacement (axialcoordinate) of the wire's end to the initial (non-deformed) length ofthe wire vs. different values of the rotation angle at the end of thewire, which experiences large (highly nonlinear) elastic deformationsunder the action of a compressive force, which is applied to the end ofthe wire in the axial direction;

FIGS. 4 a and 4 b are schematic diagrams showing wires, idealized asbeams, lying on, or embedded in, a continuous elastic or inelsticfoundation (medium) and subjected to a compressive force applied to thewires' free end;

FIG. 5 is a schematic diagram showing an initially curved wire clampedat its ends, lying on a (partial) elastic foundation and subjected tocompression;

FIG. 6 is a schematic diagram showing of an “Euler” test vehicle;

FIG. 7 depicts the mechanical behavior of a wire with an offset of itsends, clamped at its ends, and subjected to compression;

FIG. 8 is a table showing values of the applied (“interfacial”) pressurep, the ratio T/T_(e)=p/p_(e) of the actual compressive force (pressure)to its critical value, the ratios a/l, b/l, of the lateral, a, and theaxial, b, coordinates of the wire tip to the wire's initial length, l,and the ratio λ/l=1−b/l of the axial displacement λ0 to the wire'sinitial length, l computed for different a values;

FIG. 9 is a diagram showing examples of elastic curves of a deflectedwire in its pre-buckling, buckling, and immediate, i.e. close tobuckling, post-buckling condition; and

FIG. 10 is a table showing an example of data for the actual physicaldesign of a CNT array (“forest”) module.

DETAILED DESCRIPTION OF THE INVENTION

The invention disclosed herein uses the phenomena of bending and theelastic stability or instability of rods, whether free-span, i.e.un-embedded into any elastic or inelastic continuous medium, rods, orthose that can be idealized as beams lying on continuous foundations. Inaccordance with one embodiment of the invention, these phenomena areused to provide and control the required contact pressure for awire-grid-array (WGA). A key objective of the invention is to develop aWGA design that is least sensitive to the inevitable variations inmaterials' properties and wire (“beam”) geometries (diameters, lengths,prismaticity, etc.). The WGA may comprise of, but not limited to, aplurality of rods, which include, but are not limited to, nano-rods,nano-wires, carbon nanotubes (CNT), optical fibers, carbon nano-fibers,Gecko type “hair”, Velcro-type elements, and the like. For the purposeof this disclosure, the terms wire(s) or rod(s) are used interchangeablyto indicate, without limitation, nano-rods, nano-wires, carbon nanotubes(CNT), wires, beams, pillars, optical fibers, carbon nano-fibers, Geckotype “hair”, Velcro®-type elements, and the like, where a wire is anystructural element that is subjected to bending by the application of anaxial bending force, with or without actual bending of the wire. Inaccordance with the invention, a wire may or may not be partially orfully embedded within an embedding material, as discussed in more detailbelow.

Reference is now made to FIG. 1 where an array of rods, for example aCNT array, intended for heat removal, is shown. The rods 170 may be of acomposed type, with or without an initial curvature, and lateral (“endsoffset”) and/or angular misalignment of the wire ends. It has been found(see, for instance, P. Kim et al., Physical Review Letters, vol. 87,2001) that CNTs can provide extraordinary high heat transfercapabilities and therefore there is an obvious incentive to employ themin heat removing devices.

In accordance with the invention, it is disclosed that WGAs, such asvertically aligned multi-walled rods, are manufactured on silicon ornon-silicon, e.g. metal bases, silicon wafers, etc., where the rootportions of the CNTs are positioned, or otherwise anchored within asingle or multi-layer structure, having a sufficiently high effectiveYoung's modulus and a sufficiently high coefficient of thermal expansion(CTE), intended to provide a reliable anchoring of the WGA root portion,as well as better heat transfer. The structure 150 is placed inside ofthe base 120, that may further include spacers 130 that provide anadequate protrusion of the CNTs so that their axial displacements underthe action of the axial forces are not larger than necessary for theircompression-induced bending or buckling. The spacers 130 are placedbetween the base 120 and a hot body 110, the hot body being, forexample, the back side of a semiconductor chip. Hot body 110 provides acontact interface and an area of contact pressure. The WGA is furtherplaced in a low modulus continuous medium (further referred to as “lowmodulus material”, or LMM) 160, in which the rods 170 are fully, orpartially, embedded and which has, if necessary, good heat transfer andother useful mechanical and physical characteristics, such as the rightviscosity, good wettability with respect to the contact materials, etc.In addition, an optional single or multi-layer structure 140 may beneeded to provide a second support for the WGA to make its performance,as far as contact pressure is concerned, more consistent andpredictable. The rods or wires 170 protrude beyond the walls of thebase, i.e. substrate, 120 and establish contact with hot body 110 priorto establishing of contact by spacers 130 of base 120.

Reference is now made to FIG. 2, which shows a schematic of a wireexperiencing large, non-linear, elastic deformations, and to FIG. 3where a corresponding table shows values in respect of the bendingangles. Accordingly, the data in FIG. 3 enable the determination of therequired protrusion and the required axial displacement for the given(designed) pressure for the initially straight or bent wire clamped atits low end and free at the other. The critical (Euler) force isdetermined, for a wire that could be idealized as a cantilever beam, as:$\begin{matrix}{T_{e} = {\frac{\pi^{2}{EI}}{4l^{2}} = \frac{\pi^{3}{Ed}^{4}}{256l^{2}}}} & (1)\end{matrix}$

FIG. 4A, shows a schematic 400 of a wire 402 that can be idealized as abeam lying on a continuous elastic foundation. In this case the critical(Euler) force is expressed as: $\begin{matrix}{T_{e} = {{2\sqrt{KEI}} = {\frac{1}{4}d^{2}\sqrt{\pi\quad{KE}}}}} & (2)\end{matrix}$where K is the spring constant of the foundation. The critical pressureinduced by this wire is: $\begin{matrix}{p = \sqrt{\frac{KE}{\pi}}} & (3)\end{matrix}$and it is, therefore, independent of the wire length and diameter. In atypical and exemplary nanotube, the diameter of the wire is in the rangeof 20-50 nanometers, and the wire length is 20-100 micrometers. Thebuckling pressure to be applied is for a typical and exemplary nanotubein the range of 10-100 pounds per square inch (psi). In one embodimentof the invention, and as further shown in schematic 420 of FIG. 4B, thewire 422 is coated by a coat 424, the coat being, for example, of heavymetal such as copper, gold, titanium, nickel, or the like, for thebetter mechanical and, optionally better thermal performance of the WGA.The main objective of this coat, as far as the mechanical behavior ofthe wire is concerned, is to lead to a greater distributed weight of thewire 422, so that a lower force, T, is required to buckle the wire.Specifically, a coated wire is essentially closer to its buckling point,but nevertheless is still at its pre-buckling state.

In yet another embodiment of the invention, the upper portion of thecoat 424, that may be in contact with another surface (for example withthe surface of a semiconductor chip), is further coated with a liquidresistant, or barrier, material such as Teflon, to avoid direct contactbetween the wet surface of an LMM, as described in ore detail above, andthe surface of such a chip. The main objective of such a barrier, alsoreferred to as a “creep barrier”, material is to keep the LMM trappedwithin the WGA and not to allow the LMM to “creep out” the WGA. A personskilled-in-the-art would readily realize that the teachings above may beapplied separately or in any combination to achieve the desired results,without departure from the scope and spirit of the invention.

FIG. 5 shows a schematic of an initially curved wire with two endsclamped and partially supported by an elastic foundation. Notably, awire with an initial curvature, and/or ends offset, behaves as anonlinear spring, in which the axial and lateral displacements graduallyincrease with an increase in the applied axial force, while an initiallystraight wire remains straight up to the very moment of buckling. In theactual design the wire may or may not be embedded (fully or partially)into a LMM, and the LMM can be based on an organic (e.g., silicone geltype) or inorganic (e.g., Ga, In—Ga or any other low temperature meltingalloy, which is liquid at room temperature, and, or at the operatingtemperatures) material.

In accordance with the invention, an Euler test vehicle is designed forapplying a given pressure to a specimen under operation and/or test, forthe purpose of keeping the pressure within a relatively narrow anddesired, or otherwise required, limit. The mechanical behavior of thetest vehicle shown in FIGS. 6 and 7 is based on the use of thephenomenon of buckling of rods (wires). The initial curvature, and/orends offset, does not change the critical (buckling) force, but has afavorable effect on the pre-buckling behavior of the wires: in a wirewith an initial curvature (offset of its ends) the lateral and axialdeflections increase gradually when the applied axial (compressive)force changes from zero to its finite (including critical) value andacts as a nonlinear spring, while an initially straight wire remainsstraight up to the very moment of buckling, and starts exhibiting itsspring effects only in the post-buckling mode.

The principle of the operation of the device shown in FIG. 6 is madeapparent in conjunction with the drawing in FIG. 7. The pressing element600 is designed in accordance with the dimensions of the testedspecimen. The pressure is applied by rotating a screw 610. The threadsof screw 610 should be fine enough to ensure accurate axial displacementand corresponding pressure. The axial motion and applied pressure isread from a dial 630. Alternatively, a strain gage of any kind could beused to measure the induced force and, if necessary, the correspondingaxial displacement. Any non-contact way of measuring the axialdisplacements is also possible, for example, measuring the electricalcapacitance between the WGA base and the specimen under test. In oneembodiment of the invention, wires 660 are arranged around the peripheryof a test specimen 650, leaving the mid-portion of the specimen 650accessible. In yet another embodiment of the invention, the wires arearranged in accordance with the array of wires of a conceived device.The vertical structural elements 640 are of a circular cross-section.Pressure is applied to the wires 660 by means of a plate 620 that isdisplaced vertically by the turn of the screw 610. Hereinafter, there isprovided a detailed discussion of the various types of behaviorsdiscussed in general terms above.

Critical force and contact pressure for a cantilever free-span wire withan initial curvature (pre-buckling and buckling modes).

The differential equation of bending for such a wire is:EIw ^(iv)(x)+T[w ^(II)(x)+w ₀ ^(II)(x)]=0  (4)where w(x) is the deflection function, w₀(x) is the initial curvature,EI is the flexural rigidity of the wire and T is the applied axialforce. Equation (4) takes into consideration the fact that the bendingmoment produced by the compressive force T is due to both the initialcurvature of the wire and to its force-induced curvature. The totaldeflection of the wire can be found as: $\begin{matrix}{{w(x)} = {\frac{f_{0}}{1 - \frac{T}{T_{e}}}\sin\frac{\pi\quad x}{2l}}} & (5)\end{matrix}$where f_(o) is the maximum initial deflection, l is the initial lengthof the wire span, and $\begin{matrix}{T_{e} = {\frac{\pi^{2}{EI}}{4l^{2}} = {\frac{\pi^{3}}{256}{E( \frac{d^{2}}{l} )}^{2}}}} & (6)\end{matrix}$is the buckling (critical) force. The pressure produced by WGA can beeasily found as: $\begin{matrix}{p = {\frac{\pi^{2}}{64}\eta\quad{E( \frac{d}{l} )}^{2}}} & (7)\end{matrix}$where η is the portion of the total area covered by the wires,indicating the percentage of the total area of the CNT base thatactually experiences the compressive force, E is Young's modulus of thewire material, and d/l is the ratio of the wire diameter to itsrespective length.

The axial displacements can be found as: $\begin{matrix}{\lambda = {\frac{1}{l}\frac{\pi^{2}}{16}\frac{f_{0}^{2}}{( {\frac{T_{e}}{T} - 1} )^{2}}}} & (8)\end{matrix}$

The formulas 4-8 enable one to design the WGA for the required pressurein the pre-buckling mode.

Critical force and contact pressure for a cantilever free-span wire(post-buckling mode).

It is assumed that in the post-buckling mode, the initial curvature doesnot have to be considered. The distances a and b, shown in FIG. 2, canbe found as: $\begin{matrix}{{a = \frac{2p}{k}},{b = {{\frac{2}{k}{E(p)}} - l}},} & (9)\end{matrix}$where $\begin{matrix}{{p = {\sin\frac{\alpha}{2}}},{k = \sqrt{\frac{T}{EI}}},{l = \frac{K(p)}{k}},} & (10)\end{matrix}$and the functions K(p) and E(p) elliptic integral of the first and thesecond kind respectively, are tabulated. The calculated data are shownin the table in FIG. 3.

In practical calculations, one should assume certain T/T_(e) ratios andevaluate the corresponding a/l ratios and b/l ratios. Then determine thecritical force using formula (6), and the value of the actualcompressive force T≧T_(e). Then one should determine, for the a valuefrom the table in FIG. 3, the module p=sin(α/2) of the elliptic functionand the parameter k from the second formula in (10). Then one shoulddetermine the functions K(p) and E(p) from the existing mathematicaltable shown in FIG. 3. Then determine the l value from the third formulain (10). Then one should determine the a and b distances from the knowna/l and b/l ratios. The required length-to-diameter ratio can be foundfrom the formula (7) for the given pressure p as $\begin{matrix}{\frac{l}{d} = {\frac{\pi}{8}\sqrt{\eta\frac{E}{p}}}} & (11)\end{matrix}$

Critical force and contact pressure for a wire (CNT) embedded with a lowmodulus medium (pre-buckling and buckling modes).

In the case of a fully embedded wire, the buckling force of a wire of acircular cross-section can be found as: $\begin{matrix}{T_{e} = {{2\sqrt{KEl}} = {\frac{d^{2}}{4}\sqrt{\pi\quad{KE}}}}} & (12)\end{matrix}$where K is the spring constant of the elastic foundation provided by theLMM.

The induced pressure can be computed as: $\begin{matrix}{p = {2\eta\sqrt{\frac{( {1 - v} )( {3 - {4v}} ){EE}_{0}}{( {1 + v} )\lbrack {{( {3 - {4v}} )^{2}\ln\frac{r_{1}}{r_{0}}} - \frac{{r_{1}^{2}/r_{0}^{2}} - 1}{{r_{1}^{2}/r_{0}^{2}} + 1}} \rbrack}}}} & (13)\end{matrix}$where v is Poisson's ratio of the low modulus medium, and$\begin{matrix}{{\frac{r_{1}}{r_{0}} \cong \frac{{( {3 - {2v}} )( {3 - {4v}} )} + 1}{2( {1 - v} )( {3 - {4v}} )s}},} & (14)\end{matrix}$where s is a small enough number. Formulas (13) and (14) presume thatthe pressure p is produced by a LMM region that experiences appreciabledeformations due to the bent (buckled) wire.

With v=0.5, s=0.1, and v=0.25, formula (14) yields:p=0.186√{square root over (EE₀)}  (15)where E is Young's modulus of the CNT material and E₀ is Young's modulusof the embedding material, which is idealized as a low modulus medium.As one sees from the formulas (13) and (15), the induced pressuredepends only on the Young's modulus of the material and is independentof the diameter-to-length ratio of the wire. Formula (15) indicates alsothat the required (for the given contact pressure) compliance of the LMMis dependent on the Young's modulus of the WGA material.

A person skilled-in-the-art would appreciate that the disclosedinvention is not limited to a single application and, without limitingthe scope of the invention, the invention is applicable to multipleforms of rods or wires. These include, but are not limited to, wires,nanotubes, nano-rods, Gecko-type “hair”, Velcro-type elements, etc.Hence, a device and a test device for creating a predetermined(“prescribed”), predicted and controlled contact pressure, for anyapplication, by using the phenomenon of elastic stability of rods or thebending deformation of initially deflected rod (wire)-like elements, istherefore shown. Either the total lengths of the wires or only theirprotruded portions may be subjected to bending, i.e., experience elasticinstability, or buckling, as a result of the application of the axialforce.

In addition, pre-buckling, buckling and post-buckling (highly nonlinear)deformations (deflections) of rods (wires, CNTs, etc.) are considered,and the mechanical behavior of the wires is predicted with highaccuracy, whether on a deterministic or a probabilistic basis, with anemphasis on the produced contact pressure, and is further addressed. Thedevice is further designed for removing heat from a hot surface, such asa high power chip.

A device in accordance with the disclosed invention is designed forflexible (“Euler”) electrical test probes, for ferruled high-densityfiber array connectors, and/or for an “Euler” test vehicle, which isintended for providing the required lateral pressure on a specimen undertest. The wires in such a device may be obliquely oriented with respectto the base. The wires can be straight or can have a more complicatedconfiguration. Such a configuration can be achieved by providingunidirectional plasma flow and/or unidirectional air blow during thefabrication of the rods/wires. The flow can be initiated at differentpoints of time and its intensity can be changed/adjusted depending onthe particular wire configuration that is intended to be achieved.

Furthermore, a device in accordance with the invention may be designedfor the experimental evaluation of Young's modulus of a rod/wirematerial from the measured axial force and the correspondingdisplacement, and based on the developed analytical predictive modelthat enables one to evaluate the said Young's modulus from linear ornonlinear elastic deformations of the wire. This can be achieved bymeasuring the axial force and the axial displacement, by solving theequation for the Euler force for the Young's modulus of the material,and by computing this modulus for the measured force and thedisplacement for different load levels. This technique can reveal alsoany nonlinear relationship, if any, between the force (stress) and thedisplacement (strain).

In another embodiment of the disclosed device cantilever(“clamped-free”, i.e., clamped at one end and free at the other)rods/wires are used. In some other cases (in the case of, say,“clamped-clamped” wires), both ends of the rods/wires are supported.Specifically, while the root end of the wire is rigidly clamped (firmlyanchored), the upper end, although restricted from angular deformations,can slide in the axial direction under the action of a compressive forceapplied to its tip. The angles of rotation at both ends are supposed tobe zero or next-to-zero, if the support is not ideally rigid, during thewire deformation. In yet another embodiment of the invention the initialcurvature (and/or the ends offset) is predefined, introduced andconsidered in the pre-, post- or buckling modes of wire performance.Furthermore, the geometry of the wires (length, diameter, initialcurvature, ends offset) may be predefined (established beforehand), andtheir mechanical behavior can be predicted with high accuracy, based onthe developed analytical models. The wires may be further coated in oneor more layers for improved mechanical behavior in bending and buckling.

In yet another embodiment of the invention the root portions of the rodsare reliably anchored using, if necessary, high modulus, high CTE, and,in the case of thermal devices, also highly thermally conductivematerials. These materials can be applied onto the wire grid array(WGA), for example, by chemical vapor deposition, physical vapordeposition, plasma deposition, ion sputtering, electrochemicaldeposition, by casting from liquid phase, or in any other suitable way.The employment of a LMM (whose matrix can be based on a silicone geltype material; highly viscous and, if necessary, thermo-conductivefluid; thermal grease, including phase changing materials;low-temperature melting metals, such as, e.g. Ga and In—Ga alloys,etc)., enables one to convert a free-span wire to a wire whosemechanical behavior can be idealized as those for a beam lying on acontinuous elastic foundation. The employment of a LMM can be needed forlesser sensitivity of the contact pressure to the variations in the wireparameters. The resulting pressure is wire length and its diameterindependent and depends only on the Young's modulus of the wire materialand the spring constant of the elastic foundation (embedding medium). Inaddition, the employment of a LMM provides reliable separation betweenrods, thereby avoiding direct contact of the rods with each other (a“spaghetti” situation) and further stabilizing the mechanical behaviorof the WGA. The LMM may have a high thermal conductivity if such isnecessary, as well as adequate viscosity, good wettability, and otherphysical and chemical attributes, if such properties are also importantfor successful manufacturing and performance of the module.

In yet another embodiment of the invention there is employed a highmodulus, low CTE material for providing a secondary support for theupper end of long enough rods, if such a support is deemed to benecessary for a more consistent mechanical and/or functionalperformance. Notably, in accordance with the invention, partially ortotally embedded wires with constant or variable mechanicalcharacteristics of the embedding medium, are also possible. The endportions of the wires may be further embedded into a low modulus andhighly thermally conductive material that is filled with various highlyconductive particles, including nano- and/or micro-particles and CNTfibers. The device may be attached to the hot body by using the Geckophenomenon and/or Velcro concept and/or vacuum air caps, or in any otherway.

It is further noted that van der Walls forces of different nature may beresponsible for the Gecko effect and are further enhanced due to theapplication of an external electric field. While Gecko wires supposedlyhave only orientation-related forces, the induced and dispersion Van derWalls forces can be used, in addition to the orientation forces, toenhance the Gecko effect.

The disclosed device may be designed on the basis of a probabilisticapproach, which takes into consideration various inevitableuncertainties and variabilities of the materials and geometriccharacteristics of the WGA and is used to ensure that the probability tofind the contact pressure within the required range is sufficientlyhigh. The probabilistic approach can be used also to revisit the designand materials selection, if the predicted probability of a functionaland/or materials failure is higher than the acceptable level.

A person skilled-in-the-art would realize that the benefits of thedisclosed invention may be easily adaptable for use in engineeringapplications well beyond the described applications, in the fields ofengineering such as civil, electrical, mechanical, ocean, aerospace,automotive, and structural engineering, to name but a few. Therefore,the disclosure herein above should be read to be equally useful in otherareas of engineering, without limitations of the specific examplesprovided herein.

The analysis hereinafter is provided for the purpose of further teachingrelated to the above invention and assisting in the understanding of theconsiderations taken in practicing the invention. The analysis is notintended to limit the invention, but rather provide a certaintheoretical basis for the invention hereinabove. Specifically, flexiblewire-like structural elements subjected to axial compression andexperiencing bending deformations in the pre-buckling, buckling orpost-buckling modes (“Euler” rods) have been used to provide a desirablespring effect in some mechanical, electrical and optical devices. In theanalysis that follows we model the mechanical behavior of cantileverwire-like elements subjected to axial compression having in mindprimarily thermal interfaces that employ carbon nano-tube (CNT) arrays(“forests”). The obtained formulas enable one to design a CNT or anyother wire-grid array (WGA) with the prescribed, predicted andcontrolled interfacial (contact) pressure. They also enable one topredict the configurations (elastic curves) of the wires. The contactpressure should be high enough for the adequate performance of thedevice, but low enough not to compromise the strength of the contactmaterials. We intend to examine free span wires, as well as wiresembedded into an elastic medium. By embedding a WGA into a LMM one canmake the contact pressure insensitive to variations in the geometriccharacteristics of the WGA. We suggest that, if possible, the WGA isdesigned in such a way that the wires operate in buckling and/orslightly post-buckling conditions. In such a case, the induced pressureis least sensitive to variations in the axial displacements and remainsclose to the buckling pressure.

The analysis performed is based on the following assumptions:

-   -   a) Wire-like structural elements in question can be idealized as        cantilever beams (flexible rods/wires), and the engineering        theory of bending and buckling of beams can be used to analyze        their mechanical behavior;    -   b) Structural elements in question perform elastically for any        stress/strain level;    -   c) Stress-strain relationship for the WGA material is linear,        i.e. stress/strain level independent;    -   d) WGA wires are characterized by large length-to-diameter        ratios, so that the shearing deformations need not be accounted        for; this is generally true, as long as these ratios are larger        than, say, 12-15;    -   e) Hollows in wire structures, if any, are small, do not affect        their mechanical behavior and need not be considered;    -   f) Wires are rigidly clamped at their root cross-sections and        free at the other ends;    -   g) Contact surface (interface) is ideally smooth and does not        restrict the lateral displacements of the wire tips at any        strain/displacement level;    -   h) Wire-like structural elements embedded into a LMM can be        treated, or otherwise idealized, as flexible beams lying on a        continuous elastic foundation provided by such a medium;    -   i) Spring constant of the elastic foundation provided by the LMM        should be evaluated, however, for a wire-like element embedded        into, but not actually lying on, an elastic medium, i.e. by        considering the fact that the wire is surrounded by, and is        attached to, the LMM;    -   j) Elastic modulus of the wire material is significantly higher        than the modulus of the embedding medium, and therefore the        circumferential shape of the wire cross-section remains circular        during the wire deformations; in other words, the spring        constant of the elastic foundation can be evaluated assuming        that the cross-section of the wire is displaced, but is not        deformed; and    -   k) Initial curvatures (deflections) of the wire-like elements        affect the pre-buckling (linear) behavior of the wire, but do        not affect the large post-buckling deflections of the wires.

Following there is provided an analysis of free-span wires, beginningwith the behavior in the pre-buckling mode.

The equation of bending (equilibrium) for a wire-like structural elementsubjected to axial compression is as follows:EIw ^(IV) (x)+T[w ^(II)(x)+w ₀ ^(II)(x)]=0.  (16)

Here w(x) is the (force-induced) deflection function, w₀ (x) are theinitial (stress free) deflections that can be represented as$\begin{matrix}{{{w_{0}(x)} = {f_{0}\sin\frac{\pi\quad x}{2l}}},} & (17)\end{matrix}$where EI is the flexural rigidity of the wire, E is the Young's modulusof the wire material, I=πd⁴/64 is the moment of inertia of the wirecross-section, d is its diameter, l is the wire length, f₀ is themaximum initial deflection (at the wire tip), and T is the compressiveforce. The origin, 0, of the coordinates x, y is at the free end of thewire. The axis x is perpendicular to the clamped cross-section, and theaxis y is parallel to this cross-section.

The approximate solution to the equation (16) in the form (17) is soughtas: $\begin{matrix}{{{w(x)} = {C\quad\sin\frac{\pi\quad x}{2l}}},} & (18)\end{matrix}$where C is the constant of integration. Expression (17) and its solution(18) are consistent with the boundary conditions:w(0)=0, w ^(II) (0)=0, w ^(I)(l)=0, w ^(II) (l)=0  (19)

These conditions indicate that the deflection and the bending moment atthe wire's free end at the given force level are zero, and so are therotation angle and the lateral force at the wire's root cross-section.Introducing the formulas (17) and (18) into the equation (16) we findthat: $\begin{matrix}{{C = \frac{f_{0}}{\frac{T_{e}}{T} - 1}},} & (20)\end{matrix}$where $\begin{matrix}{T_{e} = {\frac{\pi^{2}{EI}}{4l^{2}} = {\pi^{3}{E( \frac{d^{2}}{16l} )}^{2}}}} & (21)\end{matrix}$is the critical (Euler) force for a cantilever beam. Then the solution(18) results in the following formula for the force-induced deflection:$\begin{matrix}{{w(x)} = {\frac{f_{0}}{\frac{T_{e}}{T} - 1}\sin\frac{\pi\quad x}{2l}}} & (22)\end{matrix}$

The total deflection is $\begin{matrix}{{w_{t}(x)} = {{{w_{0}(x)} + {w(x)}} = {\frac{f_{0}}{1 - \frac{T}{T_{e}}}\sin\frac{\pi\quad x}{2l}}}} & (23)\end{matrix}$

The axial force-induced displacement of the wire tip can be found as$\begin{matrix}{{\lambda = \frac{\lambda_{0}}{( {\frac{T_{e}}{T} - 1} )^{2}\quad}},} & (24)\end{matrix}$where $\begin{matrix}{\lambda_{0} = {( \frac{\pi}{4} )^{2}\frac{f_{0}^{2}}{l}}} & (25)\end{matrix}$is the initial (stress-free) displacement. From (24) it can be easilyfound that: $\begin{matrix}{{T = \frac{T_{e}}{1 + \sqrt{\frac{\lambda_{0}}{\lambda}}}},{p = \frac{p_{e}}{1 + \sqrt{\frac{\lambda_{0}}{\lambda}}}},} & (26)\end{matrix}$where $\begin{matrix}{p = \frac{4T}{\pi\quad d^{2}}} & (27)\end{matrix}$is the actual contact pressure and $\begin{matrix}{p_{e} = {\frac{4T_{e}}{\pi\quad d^{2}} = {E( \frac{\pi\quad d}{8\quad l} )}^{2}}} & (28)\end{matrix}$is its critical value.

The following conclusions could be drawn from the above analysis:

-   -   a) Within the framework of the taken approach, the actual force        (pressure) reaches half of its critical value, when the        force-induced axial displacement becomes equal to the initial        axial displacement;    -   b) A linear approach predicts accurately the critical force        (pressure), while the formulas for the predicted displacements        are valid only for displacements that are considerably lower        than their critical values;    -   c) While the total deflection can be obtained as a sum of the        initial deflection and the force-induced deflection, the total        axial displacement cannot be found in the same fashion; this is        because the axial compliance (spring constant) is displacement        dependent, i.e. the force-displacement relationship is not        linear;    -   d) The actual and the critical forces are proportional to the        wire diameter to the fourth power (because they are proportional        to the moment of inertia of the cross-section), and are        therefore sensitive to the change in this diameter. The        corresponding pressures, however, are proportional to the        diameter-to-length ratio squared, and, for this reason, are less        sensitive to the variability in the wire diameter; and    -   e) It is the diameter-to-length ratio that defines the pressure        level, and not the diameter and the length taken separately.

The axial compliance of the wire can be found as the ratio of the axialdisplacement of the wire tip to the corresponding compressive force asfollows: $\begin{matrix}{C = {\frac{\lambda}{T} = {\frac{\lambda_{0}}{{T( {\frac{T_{e}}{T} - 1} )}^{2}} = \frac{\sqrt{\lambda}( {\sqrt{\lambda} + \sqrt{\lambda_{0}}} )}{T_{e}}}}} & (29)\end{matrix}$

This formula indicates that indeed the initial deflection and the axialdisplacement need not be considered if the force-induced axialdisplacement λ is substantially larger than the initial axialdisplacement λ₀. This takes place, when the wire performs in thenear-buckling, buckling or post-buckling mode. In such a situation, i.e.when the initial displacement is either zero or need not be accountedfor, the wire compliance can be evaluated as $\begin{matrix}{C = \frac{\lambda}{T_{e}}} & (30)\end{matrix}$

Let, for instance, a CNT array (“forest”) is used as a suitable WGA.Assuming that the Young's modulus of the CNT material is as high as:E=1.28 TPa=1.306×10⁵ kg/mm²=185.7 Mpsi⁶,putting CNT length and diameter as l=40 μm, d=40 nm, and using theformula (21), it can be easily found that the critical force isT_(e)=2.531×10⁻¹¹ kg. Let, for instance, only 40% of the contact area becovered by the WGA. Then the predicted contact interfacial pressure dueto the buckled CNTs is: $\begin{matrix}{p_{e} = {{\eta\frac{4T_{e}}{\pi\quad d}} = {{\eta\quad{E( \frac{\pi\quad d}{8\quad l} )}^{2}} = {{{8.056 \times 10^{- 3}}{kg}\text{/}{mm}^{2}} = {11.5\quad{psi}}}}}} & (31)\end{matrix}$

The post-buckling behavior of the wire can be evaluated, for largedeflections, using the “elastica” model. The initial length, l, of thewire, the ratio T/T_(e)=p/p_(e) of the actual force (pressure) to itscritical value, and the coordinates y=a (parallel to the clampedcross-section) and x=b (perpendicular to the clamped cross-section) ofthe wire's root end with respect to the origin (located at the wire tip)are as follows: $\begin{matrix}{{l = \frac{K(p)}{k}},{\frac{T}{T_{e}} = \lbrack {\frac{2}{\pi}{K(p)}} \rbrack^{2}},{a = \frac{2p}{k}},{b = {{\frac{2}{k}{E(p)}} - l}},} & (32)\end{matrix}$where, $\begin{matrix}{k = {\sqrt{\frac{T}{El}} = {\frac{8}{d^{2}}\sqrt{\frac{T}{\pi\quad E}}}}} & (33)\end{matrix}$is the parameter of the axial force, p=sin(α/2) is the module of theelliptic function, α is the angle of rotation at the free end, i.e. theangle that the tangent to the wire's elastic curve at its free end formswith the x axis, and $\begin{matrix}{{{K(p)} = {\int_{0}^{\pi/2}\quad\frac{\mathbb{d}\quad\phi}{\sqrt{1 - {p^{2}\sin^{2}\phi}}}}},{{E(p)} = {\int_{0}^{\pi/2}{\sqrt{1 - {p^{2}\sin^{2}\phi}}{\mathbb{d}\phi}}}}} & (34)\end{matrix}$are the tabulated complete elliptic integrals of the first and thesecond kind, respectively. The values of p, T/T_(e)=p/p_(e), a/l, b/l,and λ/l=1−b/l computed for different α values are shown in FIG. 8. Let,for instance, the force-induced axial displacement, λ, be λ=1.208 μm, sothat, with l=40 μm, we have λ/l=0.0302. Then the data shown in FIG. 8predict that p=1.01534p_(e)=1.01534×11.5=11.7 psi.

It is advisable, whenever possible, to reach and, preferably, even tosomewhat exceed the critical pressure, because in the neighborhood ofthe critical condition the actual pressure, as evident from Table 1data, is not very sensitive to the change in the axial displacement. Onthe other hand, the sensitivity analysis based on the second formula in(26) shows that when the actual displacement is small and, as a resultof that, the contact pressure is low, this pressure rapidly increaseswith an increase in the axial displacement. In the above example, thea/l and the b/l ratios are 0.2194 and 0.9698, respectively, so that thecoordinates of the wire tip with respect to its root cross-section area=8.78 μm and b=3.88 μm. The information of the induced displacements isuseful, particularly, for the selection of the adequate average spacingbetween the adjacent wires, so that to avoid a “spaghetti” situation,when the deflected wires touch each other and form a bundle, instead ofan array, clearly undesirable from the standpoint of a consistent andpredictable mechanical behavior of the WGA.

The axial compliance can be found as the ratio of the axialforce-induced displacement to the corresponding axial force:$\begin{matrix}{C = {\frac{\lambda}{T} = {\frac{\lambda}{T_{e}}\frac{T_{e}}{T}}}} & (35)\end{matrix}$

Comparing this formula with the formula (30), we conclude that thefactor T_(e)/T reflects the effect of the large axial displacements ofthe wire tip on the wire's axial compliance. The axial induceddisplacement can be evaluated as $\begin{matrix}{\lambda = {{l - b} = {{2l} - {\frac{2}{k}{E(p)}}}}} & (36)\end{matrix}$

Then the formula (35) yields: $\begin{matrix}{{C = {\frac{2l^{3}}{EI}{\chi(p)}}},} & (37)\end{matrix}$where the function $\begin{matrix}{{\chi(p)} = \frac{{K(p)} - {E(p)}}{K^{3}(p)}} & (38)\end{matrix}$considers the effect of the large deflections on the axial compliance.This function is tabulated in the table shown in FIG. 8. It is zero forzero axial displacement of the wire tip and increases with an increasein this displacement.

Following is a probabilistic approach to the analysis. The analysisbelow should be used merely as an illustration of what can be expectedfrom the application of a probabilistic approach. As evident from thesecond formula in (26) and the formula (28), two major factors affectthe contact pressure in the pre-buckling mode (provided that the Young'smodulus of the material is known with sufficient certainty): the initialaxial displacement λ₀, which depends on the initial curvature (initiallateral displacement), and the diameter-to-length ratio d/l. The initialcurvature and the initial axial displacement are seldom known withcertainty and therefore should be treated as random variables. This isespecially true for CNT arrays. The diameter-to-length ratio for a CNTin a CNT array is typically not known with sufficient certainty, andshould be treated, for this reason, as a random variable. In theanalysis that follows we address, as an illustration, the effect of theinitial curvature (axial displacement). If both the variables λ₀ and d/lare considered, a cumulative probability distribution function should beobtained for these two variables.

Let the maximum initial lateral displacement f₀ be a random variable,with the mean value <f₀>, variance D_(f0) and the most likely value f*.It is natural to assume that the variable f₀ is distributed inaccordance with Weibull's law. Indeed, the sign of the displacement f₀does not matter and therefore this displacement can be assumed positive.Zero value of such a displacement is physically possible, but theprobability of such a situation is zero; and, finally, low values of thevariable f₀ are more likely than high values.

The probability that the actual initial maximum deflection exceeds acertain level f₀ is: $\begin{matrix}{{{F_{f}( f_{0} )} = {\exp\lfloor {- ( \frac{f_{0}}{b} )^{a}} \rfloor}},} & (39)\end{matrix}$where the shape parameter, a, and the scale parameter, b, are related tothe probabilistic characteristics of the random variable f₀ as follows:$\begin{matrix}{{\langle f_{0} \rangle = {{b\quad{\Gamma( {3/2} )}} = {b\frac{\sqrt{\pi}}{2}}}},{D_{f_{0}} = {b^{2}\lbrack {{\Gamma( {1 + {2/a}} )} - {\Gamma^{2}( {1 + {1/a}} )}} \rbrack}},{f_{*} = {b( {1 - {1/a}} )}^{1/a}}} & (40)\end{matrix}$where $\begin{matrix}{{\Gamma(\alpha)} = {\int_{0}^{\infty}{x^{a - 1}\quad{\mathbb{e}}^{- x}{\mathbb{d}x}}}} & (41)\end{matrix}$is the gamma-function. From (25) and (26) we find: $\begin{matrix}{f_{0} = {\frac{4}{\pi}\sqrt{\lambda\quad l}( {\frac{p_{e}}{p} - 1} )}} & (42)\end{matrix}$

Substituting this formula into the formula (39), we conclude that theprobability that the actual contact pressure does not exceed a certainlevel P (which, in this analysis, is always below its critical valueP_(e)) is: $\begin{matrix}{{F_{P}(P)} = {\exp\lbrack {- \lbrack {\frac{4}{\pi}\frac{\sqrt{\lambda\quad l}}{b}( {\frac{p_{e}}{p} - 1} )} \rbrack^{a}} \rbrack}} & (43)\end{matrix}$

Then the probability that the contact pressure is found within theboundaries P₁ and P₂ is $\begin{matrix}{P = {{\lbrack {P_{1} \leq p \leq P_{2} \vartriangleleft p_{e}} \rbrack{\exp\lbrack {- \lbrack {\frac{4}{\pi}\sqrt{\frac{\lambda l}{b}}( {\frac{p_{e}}{P_{2}} - 1} )} \rbrack^{a}} \rbrack}} - {\exp\lbrack {- \lbrack {\frac{4}{\pi}\frac{\sqrt{\lambda\quad l}}{b}( {\frac{p_{e}}{P_{1}} - 1} )} \rbrack^{a}} \rbrack}}} & (44)\end{matrix}$

In a special case of a Rayleigh distribution (a=2) we have:$\begin{matrix}{P = {\lbrack {P_{1} \leq p \leq P_{2} \vartriangleleft p_{e\quad}} \rbrack = {{\exp( {- \frac{P_{1}}{\langle p \rangle}} )} - {\exp( {- \frac{P_{2}}{\langle p \rangle}} )}}}} & (45)\end{matrix}$where

p

is the mean value of the contact pressure. This value is related to thevariance, D_(P), as follows $\begin{matrix}{\langle p \rangle = \sqrt{\frac{D_{P}}{\frac{4}{\pi} - 1}}} & (46)\end{matrix}$

Let the mean value of the contact pressure be

p

=p_(e)=11.5 psi. Then formula (45) predicts that the probability thatthe actual pressure is found within the boundaries of, i.e. P₁=5 psi andP₁=25 psi is p=0.6474-0.1137=0.5337=53%.

Following is now an analysis of the critical pressure of a wire embeddedinto an elastic medium. From the standpoint of structural analysis, awire embedded into a LMM can be treated as a cantilever beam lying on acontinuous elastic foundation provided by such a medium. It isimportant, however, that the spring constant of this foundation isevaluated for the case of a wire embedded into, and not actually lyingon, an elastic medium.

The equation of bending, with consideration of the initial curvature, isas follows:EIw ^(IV)(x)+T[w ^(II)(x)+w ₀ ^(II)(x)]+Kw(x)=0,  (47)where K is the spring constant of the foundation. The origin of thecoordinate x is at the clamped end of the wire. The initial curvaturecan be represented as $\begin{matrix}{{{w_{0}(x)} = {\frac{f_{0}}{2}( {1 - {\cos\frac{\pi\quad x}{l_{0}}}} )}},} & (48)\end{matrix}$where f₀ is the maximum initial deflection, and l₀ is the buckled lengthof the wire. The expression (33) satisfies the boundary conditions:w ₀ (0)=0, w ₀ ^(I)(0)=0, w ₀ (l ₀)=f ₀ , w ₀ ^(III)(l ₀)=0  (49)

The solution to equation (47) in the form of the initial curvature (48)is: $\begin{matrix}{{w(x)} = {C( {1 - {\cos\frac{\pi\quad x}{l_{0}}}} )}} & (50)\end{matrix}$

This solution also satisfies the boundary conditions (49). Afterintroducing the expression (48) and the sought solution (50) intoequation (47), and using Galerkin's method to solve this equation, weobtain: $\begin{matrix}{{C = \frac{f_{0}}{{\frac{EI}{T}\xi} + {\frac{K}{T}\frac{1}{\xi}} - 1}},} & (51)\end{matrix}$where the following notation is used:ζ=(π/l ₀)²  (52)

Then the solution (50) results in the following formula for the elasticcurve: $\begin{matrix}{{w(x)} = {\frac{f_{0}}{{\frac{EI}{T}\xi} + {\frac{K}{T}\frac{1}{\xi}} - 1}( {1 - {\cos\frac{\pi\quad x}{l_{0}}}} )}} & (53)\end{matrix}$

Within the framework of the taken approach, which is similar to theapproach employed above in the analysis of a free span wire, thedeflections tend to infinity, when the compressive force tends to itscritical value, T_(e). Equating the denominator in the expression (53)to zero and putting the compressive force T equal to T_(e), thefollowing formula for this force is obtained: $\begin{matrix}{T_{e} = {{{EI}\quad\xi} + \frac{K}{\xi}}} & (54)\end{matrix}$

The minimum value:T_(e)=2√{square root over (KEI)}  (55)of this force takes place for $\begin{matrix}{\xi = \sqrt{\frac{K}{EI}}} & (56)\end{matrix}$i.e. for the buckling length $\begin{matrix}{l_{0} = {\pi\sqrt[4]{\frac{EI}{K}}}} & (57)\end{matrix}$

For large enough K values, this buckling length can be considerablylower than the wire length. Hence, this explains the reason that thecritical force determined by the formula (55) is wire lengthindependent.

The critical pressure $\begin{matrix}{{{p_{e}\frac{4}{\pi\quad d^{2}}T_{e}} = \sqrt{\frac{KE}{\pi}}},} & (58)\end{matrix}$is independent of both the wire length and its diameter. Comparing theformula (58) with the formula (28), leads to the conclusion that whilethe critical pressure in the case of a free span wire is proportional tothe diameter-to-length ratio squared and is proportional to the Young'smodulus of the wire material, the critical pressure in the case of awire embedded into a LMM is independent of the wire length and diameter,and is proportional to the square root of the wire material modulus.This circumstance provides an incentive for using wires embedded into alow modulus medium, as far as the sensitivity of the critical pressureto the change in the wire geometry and Young's modulus of the wirematerial is concerned.

If, for instance, a contact pressure of 60 psi is desirable, then thespring constant of the elastic foundation provided by the embeddingelastic medium should be, with E=1.28 TPa=130612 kg/mm²=1.857×10⁸ psi,as low as K=6.09×10⁻⁵ psi. For the contact pressure of 100 psi therequired spring constant is K=1.69×10⁻⁴ psi.

It is noteworthy that each term in the right part of the formula (54)contributes half of the value of the critical force expressed by theformula (55). This means that both the flexural rigidity of the wire andthe spring constant of the elastic medium are equally important from thestandpoint of converting the buckling mode characterized by the formula(21), in the case of a free span wire, to the mode characterized by theformula (55), in the case of a wire embedded into a low modulus medium.

Equating the formulas (28) and (55) and solving the obtained equationfor the spring constant K, we conclude that the K value should not belower than $\begin{matrix}{{K_{*} = {\pi\quad{E( \frac{\pi\quad d}{8l} )}^{4}}},} & (59)\end{matrix}$if the mechanical behavior corresponding to the buckling force (55) isto be achieved. If, for instance, the CNT is characterized by adiameter-to-length ratio of 0.001, then with E=1.28 TPa=130612kg/mm²=1.857×10⁸ psi, the minimum spring constant value should be higherthan K*=1.387×10⁻⁵ psi.

Following is an analysis of spring constant of the embedding medium. Thespring constant K of the embedding medium can be evaluated by thefollowing formula obtained earlier in application to low temperaturemicrobending of dual coated silica glass fibers: $\begin{matrix}{K = \frac{{4\pi\quad{E_{0}( {1 - \nu} )}( {3 - {4\nu}} )}\quad}{( {1 + \nu} )\lbrack {{( {3 - {4\nu}} )^{2}\ln\frac{r_{1}}{r_{0}}} - \frac{( {r_{1}/r_{0}} )^{2} - 1}{( {r_{1}/r_{0}} )^{2} + 1}} \rbrack}} & (60)\end{matrix}$where E₀ is Young's modulus of the elastic medium, v is its Poisson'sratio, r₀ is the radius of the wire's cross-section, and r₁ is a largeenough radius, at which the disturbance introduced by the displaced wireto the stress field in the elastic medium becomes insignificant.Calculations indicate that the r₁/r₀ ratio in the above formula can beassumed equal to 30 for the unrestricted medium, i.e. for the situation,when the wires in a WGA are located far enough from each other andtherefore deform independently. Then the following simplification of theformula (60) can be used: $\begin{matrix}{K = \frac{4\pi\quad{E_{0}( {1 - \nu} )}}{( {1 + \nu} )( {3 - {4\nu}} )\ln\frac{r_{1}}{r_{0}}}} & (61)\end{matrix}$

Low modulus elastic media are typically characterized by Poisson'sratios close to 0.5. Then the formula (61) can be further simplified asfollows: $\begin{matrix}{K = {\frac{4\pi}{3}\frac{E_{0}}{\ln\frac{r_{1}}{r_{0}}}}} & (62)\end{matrix}$

Comparing this formula with the formula (59) we conclude that theYoung's moduli ratio E/E₀ should not be lower than $\begin{matrix}{\frac{E_{0}}{E} = {\frac{3}{4}( \frac{\pi\quad d}{8l} )^{4}\ln\frac{r_{1}}{r_{0}}}} & (63)\end{matrix}$to achieve the desirable effect due to the embedding medium. If, forinstance, l/d=1000, then, with the radii ratio r₁/r₀ equal to 30, wefind that the Young's moduli ratio should not be lower than 6.06×10⁻¹⁴.With E=1.28 TPa=1.857×10⁸ psi, we have E₀=1.125×10⁻⁵ psi.

Following is an analysis of elastic curves and a characteristicequation. We examine a situation when the cantilever wire is loaded by acompressive T and a lateral P loads applied to its free end. The lateralload can be due, for instance, to the thermal expansion (contraction)mismatch of the dissimilar materials that the wire is in contact with.An analysis based on a linear approach is employed. It can be thereforeused for displacements that do not exceed the buckling conditions. Weproceed from the following equation of wire bending:EIw ^(IV)(x)+Tw ^(II)(x)+Kw(x)=0.  (64)

The notation is the same as in the equation (16). The origin ofcoordinate x is at the clamped end of the wire. The characteristicequation for the homogeneous differential equation (64) is as follows:$\begin{matrix}{{z^{4} + {\frac{T}{EI}z^{2}} + \frac{K}{EI}} = 0} & (65)\end{matrix}$

This bi-quadratic equation has the following four roots: $\begin{matrix}{{z = {{\pm \sqrt{\frac{T}{2{EI}}}}\sqrt{{- 1} \pm \sqrt{1 - \frac{T_{e}^{2}}{T^{2}}}}}},} & (66)\end{matrix}$where the notation of equation (55) is used.

In pre-buckling mode (T

T_(e)) formula (66) can be represented as:z=±γ ₁ iγ ₂  (67)where $\begin{matrix}{{\gamma_{1} = {\frac{1}{2}\sqrt{\frac{T}{EI}}\sqrt{{- 1} + \frac{T_{e}}{T}}}},{\gamma_{2} = {\frac{1}{2}\sqrt{\frac{T}{EI}}\sqrt{1 + \frac{T_{e}}{T}}}}} & (68)\end{matrix}$

Hence, equation (64) has the following solution:w(x)=C₀ cos h γ₁ x cos γ₂ x+C₁ cos h γ ₁ x sin γ₂ x+C₂ sin h γx sin γ₂x++C₃ sin h γx cos γ₂ x  (69)where C_(n), n=0, 1, 2, 3 are constants of integration. The boundaryconditions are as follows:w(0)=0, w ^(I) (0)=0, w ^(II)(l)=0, EIw^(III)(l)+Tw^(I)(l)+P=0  (70)

The first two conditions indicate that the displacement and the angle ofrotation at the clamped end should be zero. The third condition reflectsthe fact that the bending moment at the free end should be zero. Thelast condition is, in effect, the equation of equilibrium for thefollowing forces. It is the projection of all the forces on the verticalaxis: the lateral force due to the wire deformation, the externallateral force, P, and the projection of the external compressive load T.

The solution (69) and the boundary conditions (70) result in thefollowing formulas for the constants of integration: $\begin{matrix}{{{{C_{0} = 0},{C_{1} = {{{- \frac{\gamma_{1}}{\gamma_{2}}}C_{3}} = {\frac{P}{T}\sqrt{\frac{\eta - 1}{\eta}}\sqrt[4]{\frac{4{EI}}{K}}\frac{{\sqrt{\eta^{2} - 1}\cosh\quad u\quad\cos\quad v} - {\sinh\quad u\quad\sin\quad v}}{\eta\quad D}}}}}\quad{C_{2} = {{- \frac{P}{T}}\sqrt{\frac{\eta - 1}{\eta}}\sqrt[4]{\frac{4{EI}}{K}}\frac{{\sinh\quad u\quad\sin\quad v}\quad + {\sqrt{\eta^{2} - 1}\cosh\quad u\quad\cos\quad v}}{D}}}}\quad} & (71) \\\quad & \quad\end{matrix}$where η=T_(e)/T, u=γ₁l, v=γ₂l, and $\begin{matrix}{D = {{\frac{1}{2}\cosh^{2}{u\lbrack {{( {\eta + 1} )\tanh^{2}u\quad\cos^{2}v} - {( {\eta - 1} )\sin^{2}v}} \rbrack}} + {\frac{1}{\eta}\cosh^{2}{u\lbrack {{\tanh^{2}u\quad\sin^{2}v} - {( {\eta^{2} - 1} )\cos^{2}v}} \rbrack}}}} & (72)\end{matrix}$

For a long enough wire the formulas (71) and (72) can be simplified:$\begin{matrix}\begin{matrix}{{C_{0} = 0},{C_{1} = {{\frac{\gamma_{1}}{\gamma_{2}}C_{3}} = {\frac{\quad{2P}}{T}\sqrt{\frac{\eta - 1_{4}}{\eta}}\sqrt{\frac{4{EI}}{K}}\frac{1}{\cos\quad h\quad u}\frac{{{\sqrt{{\eta^{2} - 1}\quad}\cos\quad v} - {\sin\quad v}}\quad}{( {1 + \eta} )( {2 - \eta} )}}}},} \\{C_{2} = {{- \frac{2P}{T}}\sqrt{\frac{\eta - 1_{4}}{\eta}}\sqrt{\frac{4{EI}}{K}}\frac{1}{\cos\quad h\quad u}\frac{{\sqrt{{\eta^{2} - 1}\quad}\cos\quad v} + {\sin\quad v}}{( {1 + \eta} )( {2 - \eta} )}}} \\\quad\end{matrix} & (73)\end{matrix}$

Considering the formula (57) for the buckling length, we conclude thatthe constants of integration and, hence, the ordinates of the deflectionfunction are proportional to this length. An exemplary and non-limitingelastic curve 910 for equation (69) of a deflected wire is plotted inFIG. 9 for the case:l=40 μm, d=40 nm, K=10⁻⁴ kg/mm², E=1.3509×10⁵ kg/mm²=1.32 TPa, T=0.95T_(e).Buckling Mode (T=T_(e))

In buckling mode (T=T_(e)) equation (64) has the following solution:w(x)=C₁ sin γx+C₂ γx sin γx+C₃ cos γx+C₄ cos γx,  (74)where $\begin{matrix}{\gamma = \sqrt{\frac{T}{2{EI}}}} & (75)\end{matrix}$

The boundary conditions (70) yield: $\begin{matrix}{{C_{1} = {{- C_{4}} = {\frac{2P}{T\quad\gamma}\frac{{2\quad\cos\quad u} - {u\quad\sin\quad u}}{u^{2} - 4 + {3\quad{\sin\quad}^{2}u}}}}},{C_{2} = {{- \frac{2P}{T\quad\gamma}}\frac{{\sin\quad u} + {u\quad\cos\quad u}}{u^{2} - 4 + {3\quad\sin^{2}u}}}},{C_{3} = 0},} & (76)\end{matrix}$where u=γl. An exemplary and non-limiting elastic curve 920 of equation(74) of a wire is plotted in FIG. 9. All the input data, except theaxial force, are the same as in the example for the pre-buckling mode.

In post-buckling mode (T

D T_(e)) this case the equation (64) has the following solution:w(x)=C₁ sin γ₁ x+C₂ cos γ₁ x+C₃ sin γ₂ x+C₄ cos γr₂ x  (77)where $\begin{matrix}{{\gamma_{1} = \sqrt{\frac{T}{2{EI}}( {1 + \sqrt{1 - \frac{T_{e}^{2}}{T^{2}}}} )}},{\gamma_{2} = \sqrt{\frac{T}{2{EI}}( {1 - \sqrt{1 - \frac{T_{e}^{2}}{T^{2}}}} )}},} & (78)\end{matrix}$and the constants of integration are as follows: $\begin{matrix}{{C_{1} = {{{- \frac{\gamma_{2}}{\gamma_{1}}}C_{3}} = {{- \frac{P}{EID}}( {{\gamma_{1}^{2}\cos\quad u} - {\gamma_{2}^{2}\cos\quad v}} )}}},{C_{2} = {{- C_{4}} = {\frac{P\quad\gamma_{1}}{EID}( {{\gamma_{1}\sin\quad u} - {\gamma_{2}\sin\quad v}} )}}}} & (79)\end{matrix}$whereD=γ ₁[2γ₁ ²γ₂ ²−γ₁γ₂(γ₁ ²+γ₂ ²)sin u sin v−(γ₁ ⁴+γ₂ ⁴)cos u cos v]  (80)

An exemplary and non-limiting elastic curve 930 of equation (77) isplotted in FIG. 9 for T=1.05T_(e). All the input data, except the axialforce, are the same as in the example for the pre-buckling mode.

The design for contact pressure involves the computed Young's moduli ofthe low modulus embedding medium (E₀×10⁷ psi) and length-to-diameterratios (l/d) for CNTs for the given (vs) Young's moduli of the CNT (E,TPa) and the desired/required contact pressures (P_(e), psi) are shownin FIG. 10. The table data can be used for actual physical design of aCNT array (forest) module. The length-to-diameter ratios in this tableare based on the requirement that the buckling force in the case of awire embedded into a low modulus medium is the same as in the case of afree-span wire. This gives an additional assurance that the design isstable enough, as far as the contact pressure is concerned. The datashown in FIG. 10 are based on an assumption that the entire area iscovered by the WGA. If only a portion of the array covers the interface,and/or a slightly oblique loading takes place, then the criticalpressure in the table should be multiplied by the correspondingreduction factor.

If, for instance, one wishes to achieve a constant pressure of 60 psi,and the Young's modulus of the CNT material is 1.0 Tpa, then a lowmodulus medium with the Young's modulus of 0.00006329 psi should be usedand CNTs with the length-to-diameter ratio of 611 should be employed. Ifthe diameter of the CNTs is, for example, 40 nm, then the length of theCNTs should be 24.4 microns.

The following conclusions can be drawn from the above analysis:

-   -   a) Simple formulas have been developed for the evaluation of the        contact pressure due to, and of the deformed shapes (elastic        curves) of, the bent WGA wires in WGA designs;    -   b) There is an incentive to use wires embedded in LMM for the        reduced sensitivity of the contact pressure to the geometric        characteristics of the wires;    -   c) A probabilistic analysis should be employed for the        prediction of the probabilistic characteristics of the bent        wires in WGA designs; and

The conclusions herein are not intended to limit the invention disclosedbut rather provide a theoretical analysis to the invention.

Accordingly, although the invention has been described in detail withreference to a particular preferred embodiment, those possessingordinary skill in the art to which this invention pertains willappreciate that various modifications and enhancements may be madewithout departing from the spirit and scope of the claims that follow.

1. An apparatus for the efficient removal of heat from a hot surface,the apparatus comprises: an anchoring surface wherein the root portionsof each of a plurality of wires is anchored to said anchoring surface; abase inside of which said anchoring surface is mounted; and, means forapplying axial forces to the free ends of said plurality of wires, whensaid base is forced against said hot surface; wherein said axial forcescause the bending deformation of at least one of said plurality of wiresagainst said hot surface thereby increasing at least one of: the numberof wires from said plurality of wires having thermal contact to said hotsurface, the thermal contact area between said hot surface and said atleast one of said plurality of wires.
 2. The apparatus of claim 1,wherein the contact pressure is a result of at least one of said axialforces and is only a function of the ratio between the length and thediameter of said at least one of said plurality of wires.
 3. Theapparatus of claim 1, wherein said contact pressure is controlled,predetermined, and predicted in an approach which is one of:deterministic or probabilistic.
 4. The apparatus of claim 1, theapparatus further comprises: a low modulus continuous medium (LMM), saidplurality of wires being at least partially embedded within said LMM. 5.The apparatus of claim 4, wherein said axial force is independent of thedimensions of said at least one of said plurality of wires.
 6. Theapparatus of claim 4, the apparatus further comprises: a supportstructure designed to support said plurality of wires placed in saidLMM.
 7. The apparatus of claim 6, wherein an upper portion of said atleast one of said plurality of wires is coated with a material resistantto wetting, said wetting resistant material comprising a creep barrierfor said LMM.
 8. The apparatus of claim 1, wherein said hot surface is asemiconductor device.
 9. The apparatus of claim 1, wherein asemiconductor device is mounted onto said hot surface.
 10. The apparatusof claim 2, wherein said contact pressure is in the range between 20 and60 psi.
 11. The apparatus of claim 10, wherein the ratio between thelength and the diameter of said at least one of said plurality of wiresis between 400 and 1,200.
 12. The apparatus of claim 1, wherein said atleast one of said plurality of wires having an initial curvature. 13.The apparatus of claim 1, wherein said at least one of said plurality ofwires being subjected to a lateral loading.
 14. The apparatus of claim13, wherein said lateral loading is achieved by means of coating said atleast one of said plurality of wires.
 15. The apparatus of claim 14,wherein said coating is any one of: a metal, a metal alloy, a multilayersystem, a composite structure.
 16. The apparatus of claim 15, whereinsaid metal comprises any of: copper, gold, nickel, titanium.
 17. Theapparatus of claim 14, wherein said coating is of a sufficient weight tobring said at least one of said plurality of wires up to, but less than,a buckling condition of said at least one of said plurality of wires.18. The apparatus of claim 1, wherein said at least one of saidplurality of wires having an offset.
 19. The apparatus of claim 1,wherein said wire comprises of any of: a rod, nanotube, nanowire,Gecko-type “hair”, Velcro-type element, beam, beam-like structure,fiber, and pillar.
 20. The apparatus of claim 1, further comprises: atleast partial support to the free end of at least one of said pluralityof wires.
 21. A semiconductor package that comprises an element forefficient removal of heat from the surface of a semiconductor devicemounted in said semiconductor package, the element comprises: ananchoring surface wherein the root portions of each of a plurality ofwires is anchored to said anchoring surface; a base inside of which saidanchoring surface is mounted; and, means for applying an axial forces tothe free ends of said plurality of wires, when said base is forcedagainst said hot surface; wherein said axial forces cause the bendingdeformation of at least one of said plurality of wires against said hotsurface thereby increasing at least one of: the number of wires fromsaid plurality of wires having thermal contact to said hot surface, thethermal contact area between said hot surface and said at least one ofsaid plurality of wires.
 22. The semiconductor package of claim 21,wherein the contact pressure is a result of at least one of said axialforces is only a function of the ratio between the length and thediameter of said at least one of said plurality of wires.
 23. Thesemiconductor package of claim 21, wherein said contact pressure iscontrolled, predetermined, and predicted in an approach which is one of:deterministic or probabilistic.
 24. A test device, comprising: astructural element for holding movable parts of a test probe; a screwconnected to a pressure plate, said screw having a fine thread foraccurate vertical motion of said pressure plate with respect to saidstructural element; measurement means connected to said screw, saidmeasurement means being capable of measuring at least axialdisplacement; and a bottom portion of said structural element forreceiving a test specimen, the test specimen comprising a plurality ofwires positioned to face said pressure plate; wherein said pressureplate applies a desired amount of pressure on said plurality of wires assaid screw causes said plate to move towards said plurality of wires.25. The test device of claim 24, wherein said measurement means arecapable of indicating any of pressure and displacement.
 26. The testdevice of claim 24, wherein said test specimen further comprises: ananchoring surface wherein the root portions of each of said plurality ofwires is anchored to said anchoring surface.
 27. The test device ofclaim 26, further comprising: a low modulus, continuous medium placed insaid cavity.
 28. The test device of claim 27, further comprising: asecondary support for said plurality of wires.
 29. The test device ofclaim 24, said wire comprising any of: a rod, a nanotube, a nanowire, aGecko-type “hair”, a Velcro-type element, a beam, a beam-like structure,a fiber, and a pillar.
 30. The test device of claim 24, wherein saiddesired amount of pressure is only a function of the ratio between thelength and the diameter of said at least one of said plurality of wires.31. The test device of claim 24, wherein said desired amount of pressureis controlled, predetermined, and predicted in an approach which is oneof deterministic and probabilistic.
 32. A test specimen for a testdevice designed to determine a desired amount of pressure which iscontrolled, predetermined, and predicted in an approach which is one ofdeterministic and probabilistic, wherein said test specimen is theapparatus of claim 1.